Derek Jeter may not realize this right now, and he probably would never admit it even if he drank a Big Gulp of truth serum, but the Yankees did him a favor by playing this modest version of hardball, by refusing to empty the vault for him and foisting a pay cut on him.

By agreeing to a three-year deal worth $17 million annually plus an option for a fourth year and incentives, the Yankees came up a little and Jeter came down a lot, and if the compromise landed closer to the Yankees’ target number than to Jeter’s, it will still benefit the Captain in ways he can’t possibly appreciate yet.

Because throughout a career that already has netted him over $200 million in salary, Jeter never once had been hounded by his wealth. How many athletes can say that? Any player, any sport, who breaks the bank, the bank always is there alongside him, shadowing every move he makes. Ask Amar’e Stoudemire. Ask Johan Santana. Ask CC Sabathia. Ask the patron saint of all of them, Alex Rodriguez.

Jeter? Until the past few weeks, the money he has earned has been almost incidental, which is just another charmed way that he has smartly led his professional life. It’s worth noting that many of those who have argued that Jeter deserved whatever he wanted in this negotiation have said it should be a lifetime achievement contract, a thank-you for championships already won.

Which would be fine except that most of the $189 million that Jeter banked during that freshly expired 10-year contract was in itself a pricey thank-you note. Jeter won four of his five championships before he ever signed that deal, and only won the one for the thumb in year nine of the 10-year deal. But here’s the thing: Nobody was ever bothered by that. For a man who has earned his living and his legacy on championship intangibles, nobody ever seemed cheated that those intangibles stopped delivering the ultimate bottom line after he last signed his name to a contract.

Nobody ever called him overpaid. He never had dollars tossed at him in visiting ballparks, the way Rodriguez has, even though Jeter owes that enormous deal in large part to the market-shattering $252 million standard A-Rod set shortly before Jeter’s own windfall.

Maybe that wouldn’t have bothered him. We know Jeter’s DNA, we know the way he performs every day, we have seen how unbothered he is by pressure or by criticism or by the derision of fans — like the ones in Boston who have been attacking not only his baseball talents but his manhood for 15 years. Maybe he would have laughed all of that off.

We nearly found out. Because for every Yankees fan, and every Jeter fan, that believed the Yankees were being impossibly stingy with their end of the negotiation, there are plenty of others who shed no tears over the frugality of $45 million in a bad economy. The fact that Jeter didn’t get anywhere close to his asking price may sate most of the disgruntled.

But if he had held out longer, if he had demanded the Yankees rise to the $20 million he wanted, it could have been a different story. For the first time, Jeter would have had to answer for his salary, and unless he is spending all winter in the Rejuvenator, it would have been a hard, if not impossible, task to tackle.

Now? I think most people can live with what Jeter got. And once again, it’ll be up to other Yankees — A-Rod, Sabathia, Carl Crawford perhaps, Cliff Lee for sure — to answer for their bank accounts while Jeter can simply worry about winning baseball games.